According to the official cited by the WSJ on Thursday, the training program could be modified to focus on empowering specially trained militants on the ground to call in US airstrikes.
The official added that political and military indicators justifying the move include an escalating global refugee crisis.
The issue is expected to be put to the Obama administration agenda later on Friday, when cabinet members deliberate the fight against ISIL, another official told the newspaper.
Syria has been engulfed in a civil war for over four years, as government forces battle so-called moderate opposition factions and a number of extremist groups, including ISIL.
Beginning in 2014, an international coalition led by the United States has conducted airstrikes against ISIL positions without seeking approval or support from Damascus, as the West does not recognize Assad as the legitimate leader of Syria. The coalition has been training and equipping the so-called moderate opposition.
Russia has called for cooperation between all belligerents of the Syrian civil war to prevent regional anarchy, a void that is strongly suspected would be quickly filled by ISIL.